Question

what is the actual value of , arc tan [(-1)/(1)] , arc tan [(1)/(-1)] . do they have different value? then why ? both looks like arc tan [-1] which is -pi/4 . so now what is the actual value of arc sin [(-15)/(17)] , arc tan [(15)/(-17)] . do they have different value like arc tan or do they have same value ? then why ???

Answer 1

Answer on Question #52640 – Math – Trigonometry

what is the actual value of, arc tan [(-1)/(1)], arc tan [(1)/(-1)]. do they have different value? then why? both looks like arc tan [-1] which is -pi/4. so now what is the actual value of arc sin [(-15)/(17)], arc tan [(15)/(-17)]. do they have different value like arc tan or do they have same value? then why???

Solution

Since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions. Using function in the sense of multivalued functions, the function $y = \arctan(x)$ is defined so that $\tan(y) = x$. There are multiple numbers $y$ such that $\tan(y) = x$; for example, $\tan\left(-\frac{\pi}{4}\right) = -1$, but also $\tan\left(\frac{3\pi}{4}\right) = -1$, $\tan\left(-\frac{5\pi}{4}\right) = -1$, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each $x$ in the domain the expression $\arctan(x)$ will be evaluated only to a single value, called its principal value. Principal branch for $\arctan(x)$ is $-\frac{\pi}{2} \leq y < \frac{\pi}{2}$.

Thus $\arctan(-1) = -\frac{\pi}{4}$ lies in quadrant IV where $\sin(x) < 0$, $\cos x > 0$.

Principal branch for $\arcsin(x)$ is $-\frac{\pi}{2} \leq y < \frac{\pi}{2}$.

Thus $\arcsin\left(-\frac{15}{17}\right)$ lies in quadrant IV where $\sin(x) < 0$, $\cos x > 0$.

www.AssignmentExpert.com